Then \(L^{\prime \prime \prime } = \bar{g}_2 ( \bar{g_1}^{-1} (L^{\prime } ) \cap R^{\prime }_1)\) , which is also in
\({\cal L}\) , over \((\Gamma \cup \Delta )^*\) . Note that since \(g_1\)  is a weak coding homomorphism, \(\bar{g_1}\)  is as well, and therefore \(\bar{g_1}^{-1}\)  simply operates as \(g_1^{-1}\)  does while fixing all letters of \(\Delta \) . If we then take \(L^{\prime \prime \prime }\)  and intersect it with all words where the number of \(C_i\) 's is equal to the number of \(D_i\) 's for each \(i\) , and then erase all \(C_i\) 's and \(D_i\) 's, we obtain \(\hat{L}\) . Let the Parikh image order the
letters of this alphabet
\(d_1, \ldots , d_s, C_1,D_1, \ldots , C_k,D_k\) .
This Parikh image of \(L^{\prime \prime \prime }\) 
gives a set
\(Q^{\prime \prime \prime }\subseteq \mathbb {N}_0^{s+2k}\) , which is semilinear since
\({\cal L}\)  is semilinear.
Let \(Q^{\prime \prime \prime \prime }\)  be the set obtained from \(Q^{\prime \prime \prime }\)  by
enforcing that the number of \(C_i\) 's is equal to \(D_i\) 's for each
\(i\) , which is also semilinear since the intersection of two semilinear sets is again semilinear [1]}.
Then \(\bar{Q}\) , the set obtained from \(Q^{\prime \prime \prime \prime }\)  by projection on
the first \(s\)  coordinates, is also semilinear and
this is the Parikh image of \(\hat{L}\) . Hence, \(\hat{L}\)  is semilinear.
